# Multiple-precision Reals

The `mpfr` type is based on the MPFR library. The new `mpfr` type supports correct rounding, selectable rounding modes, and many trigonometric, exponential, and special functions. A context manager is used to control precision, rounding modes, and the behavior of exceptions.

The default precision of an `mpfr` is 53 bits - the same precision as Python’s `float` type. If the precision is changed, then `mpfr(float('1.2'))` differs from `mpfr('1.2')`. To take advantage of the higher precision provided by the `mpfr` type, always pass constants as strings.

```>>> import gmpy2
>>> from gmpy2 import mpfr
>>> gmpy2.set_context(gmpy2.context())
>>> mpfr('1.2')
mpfr('1.2')
>>> mpfr(float('1.2'))
mpfr('1.2')
>>> gmpy2.get_context().precision=100
>>> mpfr('1.2')
mpfr('1.2000000000000000000000000000006',100)
>>> mpfr(float('1.2'))
mpfr('1.1999999999999999555910790149937',100)
```

The `mpfr` type supports the `__format__()` special method to allow custom output formatting.

```>>> from gmpy2 import mpfr
>>> a=mpfr("1.23456")
>>> "{0:15.3f}".format(a)
'          1.235'
>>> "{0:15.3Uf}".format(a)
'          1.235'
>>> "{0:15.3Df}".format(a)
'          1.234'
>>> "{0:.3Df}".format(a)
'1.234'
>>> "{0:+.3Df}".format(a)
'+1.234'
```

## mpfr Type

class gmpy2.mpfr(n=0, /, precision=0)
class gmpy2.mpfr(n, /, precision, context)
class gmpy2.mpfr(s, /, precision=0, base=0)
class gmpy2.mpfr(s, /, precision, base, context)

Return a floating-point number after converting a numeric value n or a string s made of digits in the given base.

A string can be with fraction-part (with a period as a separator) and/or exponent-part (with an exponent marker ‘e’ for base<=10, else ‘@’), where digits are parsed the same as the `mpz` type constructor does and both the whole number and exponent-part optionally can be preceded by ‘+’ or ‘-’.

If a precision greater than or equal to 2 is specified, then it is used. A precision of 0 (the default) implies the precision of either the specified context or the current context is used. A precision of 1 minimizes the loss of precision by following these rules:

1. If n is a radix-2 floating point number, then the full precision of n is retained.

2. If n is an integer, then the precision is the bit length of the integer.

__format__(fmt) str

Return a Python string by formatting ‘x’ using the format string ‘fmt’. A valid format string consists of:

optional alignment code:

‘<’ -> left shifted in field ‘>’ -> right shifted in field ‘^’ -> centered in field

‘+’ -> always display leading sign ‘-’ -> only display minus for negative values ‘ ‘ -> minus for negative values, space for positive values

optional width.precision

optional rounding mode:

‘U’ -> round toward plus Infinity ‘D’ -> round toward minus Infinity ‘Y’ -> round away from zero ‘Z’ -> round toward zero ‘N’ -> round to nearest

optional conversion code:

‘a’,’A’ -> hex format ‘b’ -> binary format ‘e’,’E’ -> scientific format ‘f’,’F’ -> fixed point format ‘g’,’G’ -> fixed or float format

The default format is ‘.6f’.

as_integer_ratio() tuple[mpz, mpz]

Return the exact rational equivalent of an `mpfr`. Value is a `tuple` for compatibility with Python’s `float.as_integer_ratio`.

as_mantissa_exp() tuple[mpz, mpz]

Return the mantissa and exponent of an `mpfr`.

as_simple_fraction(precision=0) mpq

Return a simple rational approximation to x. The result will be accurate to ‘precision’ bits. If ‘precision’ is 0, the precision of ‘x’ will be used.

conjugate() mpz

Return the conjugate of x (which is just a new reference to x since x is not a complex number).

digits(base=10, prec=0, /) tuple[str, int, int]

Returns up to ‘prec’ digits in the given base. If ‘prec’ is 0, as many digits that are available are returned. No more digits than available given x’s precision are returned. ‘base’ must be between 2 and 62, inclusive. The result is a three element `tuple` containing the mantissa, the exponent, and the number of bits of precision.

is_finite() bool

Return `True` if x is an actual number (i.e. non NaN or Infinity). If x is an `mpc`, return `True` if both x.real and x.imag are finite.

is_infinite() bool

Return `True` if x is +Infinity or -Infinity. If x is an `mpc`, return `True` if either x.real or x.imag is infinite. Otherwise return `False`.

is_integer() bool

Return `True` if x is an integer; `False` otherwise.

is_nan() bool

Return `True` if x is NaN (Not-A-Number) else `False`.

is_regular() bool

Return `True` if x is not zero, NaN, or Infinity; `False` otherwise.

is_signed() bool

Return `True` if the sign bit of x is set.

is_zero() bool

Return `True` if x is equal to 0. If x is an `mpc`, return `True` if both x.real and x.imag are equal to 0.

imag

imaginary component

precision

precision in bits

rc

return code

real

real component

## mpfr Functions

gmpy2.agm(x, y, /) mpfr

Return arithmetic-geometric mean of x and y.

gmpy2.ai(x, /) mpfr

Return Airy function of x.

gmpy2.atan2(y, x, /) mpfr

Return arc-tangent of (y/x); result in radians.

gmpy2.cbrt(x, /) mpfr

Return the cube root of x.

gmpy2.ceil(x, /) mpfr

Return an ‘mpfr’ that is the smallest integer >= x.

gmpy2.check_range(x, /) mpfr

Return a new `mpfr` with exponent that lies within the current range of emin and emax.

gmpy2.cmp(x, y, /) int

Return -1 if x < y; 0 if x = y; or 1 if x > y. Both x and y must be integer, rational or real. Note: 0 is returned (and exception flag set) if either argument is NaN.

gmpy2.const_catalan(precision=0) mpfr

Return the catalan constant using the specified precision. If no precision is specified, the default precision is used.

gmpy2.const_euler(precision=0) mpfr

Return the euler constant using the specified precision. If no precision is specified, the default precision is used.

gmpy2.const_log2(precision=0) mpfr

Return the log2 constant using the specified precision. If no precision is specified, the default precision is used.

gmpy2.const_pi(precision=0) mpfr

Return the constant pi using the specified precision. If no precision is specified, the default precision is used.

gmpy2.cot(x, /) mpfr

Return cotangent of x; x in radians.

gmpy2.coth(x, /) mpfr

Return hyperbolic cotangent of x.

gmpy2.csc(x, /) mpfr

Return cosecant of x; x in radians.

gmpy2.csch(x, /) mpfr

Return hyperbolic cosecant of x.

gmpy2.degrees(x, /) mpfr

Convert angle x from radians to degrees. Note: In rare cases the result may not be correctly rounded.

gmpy2.digamma(x, /) mpfr

Return digamma of x.

gmpy2.eint(x, /) mpfr

Return exponential integral of x.

gmpy2.erf(x, /) mpfr

Return error function of x.

gmpy2.erfc(x, /) mpfr

Return complementary error function of x.

gmpy2.exp10(x, /) mpfr

Return 10**x.

gmpy2.exp2(x, /) mpfr

Return 2**x.

gmpy2.expm1(x, /) mpfr

Return exp(x) - 1.

gmpy2.factorial(n, /) mpfr

Return the floating-point approximation to the factorial of n.

See `fac()` to get the exact integer result.

gmpy2.floor(x, /) mpfr

Return an `mpfr` that is the largest integer <= x.

gmpy2.fmma(x, y, z, t, /) mpfr

Return correctly rounded result of (x * y) + (z + t).

gmpy2.fmms(x, y, z, t, /) mpfr

Return correctly rounded result of (x * y) - (z + t).

gmpy2.fmod(x, y, /) mpfr

Return x - n*y where n is the integer quotient of x/y, rounded to 0.

gmpy2.frac(x, /) mpfr

Return fractional part of x.

gmpy2.frexp(x, /) tuple[int, mpfr]

Return a `tuple` containing the exponent and mantissa of x.

gmpy2.fsum(iterable, /) mpfr

Return an accurate sum of the values in the iterable.

gmpy2.gamma(x, /) mpfr

Return gamma of x.

gmpy2.gamma_inc(a, x, /) mpfr

Return (upper) incomplete gamma of a and x.

gmpy2.get_exp(x, /) int

Return the exponent of x. Returns 0 for NaN or Infinity and sets the `context.erange` flag of the current context and will raise an exception if `context.trap_erange` is set.

gmpy2.hypot(x, y, /) mpfr

Return square root of (x**2 + y**2).

gmpy2.inf(n, /) mpfr

Return an `mpfr` initialized to Infinity with the same sign as n. If n is not given, +Infinity is returned.

gmpy2.is_finite(x, /) bool

Return `True` if x is an actual number (i.e. non NaN or Infinity). If x is an `mpc`, return `True` if both x.real and x.imag are finite.

gmpy2.is_infinite(x, /) bool

Return `True` if x is +Infinity or -Infinity. If x is an `mpc`, return `True` if either x.real or x.imag is infinite. Otherwise return `False`.

gmpy2.is_regular(x, /) bool

Return `True` if x is not zero, NaN, or Infinity; `False` otherwise.

gmpy2.is_signed(x, /) bool

Return `True` if the sign bit of x is set.

gmpy2.is_unordered(x, y, /) bool

Return `True` if either x and/or y is NaN.

gmpy2.j0(x, /) mpfr

Return first kind Bessel function of order 0 of x.

gmpy2.j1(x, /) mpfr

Return first kind Bessel function of order 1 of x.

gmpy2.jn(x, n, /) mpfr

Return the first kind Bessel function of order n of x.

gmpy2.lgamma(x, /) tuple[mpfr, int]

Return a `tuple` containing the logarithm of the absolute value of gamma(x) and the sign of gamma(x)

gmpy2.li2(x, /) mpfr

Return real part of dilogarithm of x.

gmpy2.lngamma(x, /) mpfr

Return natural logarithm of gamma(x).

gmpy2.log1p(x, /) mpfr

Return natural logarithm of (1+x).

gmpy2.log2(x, /) mpfr

Return base-2 logarithm of x.

gmpy2.maxnum(x, y, /) mpfr

Return the maximum number of x and y. If x and y are not `mpfr`, they are converted to `mpfr`. The result is rounded to match the current context. If only one of x or y is a number, then that number is returned.

gmpy2.minnum(x, y, /) mpfr

Return the minimum number of x and y. If x and y are not `mpfr`, they are converted to `mpfr`. The result is rounded to match the current context. If only one of x or y is a number, then that number is returned.

gmpy2.modf(x, /)

Return a `tuple` containing the integer and fractional portions of x.

gmpy2.mpfr_from_old_binary(string, /) mpfr

Return an `mpfr` from a GMPY 1.x binary mpf format.

gmpy2.mpfr_grandom(random_state, /)

Return two random numbers with gaussian distribution.

gmpy2.mpfr_random(random_state, /) mpfr

Return uniformly distributed number between [0,1].

gmpy2.nan() mpfr

Return an `mpfr` initialized to NaN (Not-A-Number).

gmpy2.next_above(x, /) mpfr

Return the next `mpfr` from x toward +Infinity.

gmpy2.next_below(x, /) mpfr

Return the next `mpfr` from x toward -Infinity.

Convert angle x from degrees to radians. Note: In rare cases the result may not be correctly rounded.

gmpy2.rec_sqrt(x, /) mpfr

Return the reciprocal of the square root of x.

gmpy2.reldiff(x, y, /) mpfr

Return the relative difference between x and y. Result is equal to abs(x-y)/x.

gmpy2.remainder(x, y, /) mpfr

Return x - n*y where n is the integer quotient of x/y, rounded to the nearest integer and ties rounded to even.

gmpy2.remquo(x, y, /) tuple[mpfr, int]

Return a `tuple` containing the remainder(x,y) and the low bits of the quotient.

gmpy2.rint(x, /) mpfr

Return x rounded to the nearest integer using the current rounding mode.

gmpy2.rint_ceil(x, /) mpfr

Return x rounded to the nearest integer by first rounding to the next higher or equal integer and then, if needed, using the current rounding mode.

gmpy2.rint_floor(x, /) mpfr

Return x rounded to the nearest integer by first rounding to the next lower or equal integer and then, if needed, using the current rounding mode.

gmpy2.rint_round(x, /) mpfr

Return x rounded to the nearest integer by first rounding to the nearest integer (ties away from 0) and then, if needed, using the current rounding mode.

gmpy2.rint_trunc(x, /) mpfr

Return x rounded to the nearest integer by first rounding towards zero and then, if needed, using the current rounding mode.

gmpy2.root(x, n, /) mpfr

Return n-th root of x. The result always an `mpfr`. Note: not IEEE 754-2008 compliant; result differs when x = -0 and n is even. See `rootn()`.

gmpy2.rootn(x, n, /) mpfr

Return n-th root of x. The result always an `mpfr`. Note: this is IEEE 754-2008 compliant version of `root()`.

gmpy2.round2(x, n=0, /) mpfr

Return x rounded to n bits. Uses default precision if n is not specified. See `round_away()` to access the mpfr_round() function of the MPFR.

gmpy2.round_away(x, /) mpfr

Return an `mpfr` that is x rounded to the nearest integer, with ties rounded away from 0.

gmpy2.sec(x, /) mpfr

Return secant of x; x in radians.

gmpy2.sech(x, /) mpfr

Return hyperbolic secant of x.

gmpy2.set_exp(x, n, /) mpfr

Set the exponent of x to n. If n is outside the range of valid exponents, `set_exp()` will set the `context.erange` flag of the current context and either return the original value or raise an exception if `context.trap_erange` is set.

gmpy2.set_sign(x, s, /) mpfr

If s is `True`, then return x with the sign bit set.

gmpy2.sign(x, /) int

Return -1 if x < 0, 0 if x == 0, or +1 if x >0.

gmpy2.sinh_cosh(x, /)

Return a `tuple` containing the hyperbolic sine and cosine of x.

gmpy2.trunc(x, /) mpfr

Return an `mpfr` that is x truncated towards 0. Same as x.floor() if x>=0 or x.ceil() if x<0.

gmpy2.y0(x, /) mpfr

Return second kind Bessel function of order 0 of x.

gmpy2.y1(x, /) mpfr

Return second kind Bessel function of order 1 of x.

gmpy2.yn(x, n, /) mpfr

Return the second kind Bessel function of order n of x.

gmpy2.zero(n, /) mpfr

Return an `mpfr` initialized to 0.0 with the same sign as n. If n is not given, +0.0 is returned.

gmpy2.zeta(x, /) mpfr

Return Riemann zeta of x.

gmpy2.get_max_precision() int

Return the maximum bits of precision that can be used for calculations. Note: to allow extra precision for intermediate calculations, avoid setting precision close the maximum precision.

gmpy2.get_emax_max() int

Return the maximum possible exponent that can be set for `mpfr`.

gmpy2.get_emin_min() int

Return the minimum possible exponent that can be set for `mpfr`.

gmpy2.copy_sign(x, y, /) mpfr

Return an `mpfr` composed of x with the sign of y.

gmpy2.can_round(b, err, rnd1, rnd2, prec, /) bool

Let b be an approximation to an unknown number x that is rounded according to rnd1. Assume the b has an error at most two to the power of E(b)-err where E(b) is the exponent of b. Then return `True` if x can be rounded correctly to prec bits with rounding mode rnd2.

gmpy2.free_cache() None

Free the internal cache of constants maintained by MPFR.